be a locally finitely presented additive category, and let
be a finitely presented pure-injective
. We prove that
has an indecomposable decomposition if and only if every pure epimorphic
is pure-injective if and only if the endomorphism ring of
is semiperfect. This extends a
module-theoretic result which generalises the classical Osofsky Theorem.
Key Words: Locally finitely presented category, Krull-Schmidt category, indecomposable decomposition,
(completely) pure-injective object, semiperfect ring, semisimple ring, Osofsky theorem.
2010 Mathematics Subject Classification: Primary 18E05,
Secondary 18C35, 16D90.
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